7. RESULTADOS Y ANÁLISIS
7.1 CON LA MATRIZ DE ANÁLISIS
7.1.2 Resultados con las Guías de Análisis y Talleres
To gain a better understanding of the observed line ratios, we simulate the effects of resonant scattering on the radial intensity profiles of the strongest X-ray emission lines in NGC 4636. For this we use the deprojected temperature and density profiles measured with Chandra.
In our deprojection analysis, we follow the approach described by Churazov et al. (2008). We assume spherical symmetry, but make no specific assumption about the form of the underlying gravitational potential. For a given surface brightness profile in na annuli,
we choose a set ofns (ns ≤na) spherical shells with the inner radii r(i), i = 1, . . . , ns. The
gas emissivityE is assumed to be uniform inside each shell, except for the outermost shell, where the gas emissivity is assumed to decline as a power law of radius: E = Eoutr−6βout,
where βout is a parameter. The deprojection process is a simple least square solution to
determine the set of emissivities in the set of shells (along with the emissivity normalization
Eout of the outer layers) that provides the best description of the observed surface bright-
ness. The emissivity of each shell can then be evaluated as an explicit linear combination of the observed quantities. Since the whole procedure is linear, the errors in the observed quantities can be propagated straightforwardly. With our definition of η (see subsection 2.3.2), the projection matrix does not depend on energy. Therefore, the deprojection in
any energy band is precisely that used to deproject the surface brightness. Thus, we can accumulate a set of spectra (corrected for background and readout) for each of the na
annuli, and apply the deprojection to determine the emissivities of each shell in each of the ACIS energy channels.
The Chandra data were modeled using XSPEC V12 (Arnaud, 1996) and the APEC model (Smith et al., 2001). The gas temperature and normalization were free parameters in the model. The heavy element abundance was either a free parameter in the fit or fixed to 0.68 Solar (as determined from a spectral fit to Chandra data extracted within the radius of 1′ from the centre of NGC 4636). The metallicity profile in NGC 4636 has
large uncertainties, because the metal abundance distribution in the deprojected Chandra
spectra has a large scatter as a result of noise enhancement during deprojection coupled with the limited spectral resolution of the CCDs.
The electron density profile was derived from the spectral normalization, fixing the proton to electron ratio to 0.83. Fig. 2.6 shows the deprojected electron density (lower panel) and the temperature (upper panel) as a function of distance from the centre of
2 .5 M o d e lli n g o f re so n a n t sc a tt e ri n g in NG C 4 6 3 6 3 3
Table 2.3: Oscillator strengths and optical depths for the strongest X-ray lines in the spectrum of NGC 4636 for Mach numbers 0.0, 0.25, 0.5, 0.75. The expected suppression due to resonant scattering (I/I0)circ gives the value integrated
within a radius of 2 kpc from the centre of the galaxy. The approximate values (I/I0)RGS were obtained by integrating
within an “effective extraction region” which is 0.5′ wide and 3′ long. The metallicity is assumed to be constant with the
radius.
Ion λ(˚A) f M = 0.0 M = 0.25 M = 0.5 M = 0.75
τ (I/I0)circ(I/I0)RGS τ (I/I0)circ(I/I0)RGS τ (I/I0)circ(I/I0)RGS τ (I/I0)circ(I/I0)RGS
Fexvii 15.01 2.73 8.8 0.47 0.69 3.6 0.59 0.79 1.9 0.68 0.85 1.3 0.76 0.89
Fexvii 17.05 0.12 0.5 0.89 0.95 0.2 0.95 0.98 0.1 0.97 0.99 0.07 0.98 0.99
Fexviii 14.20 0.57 1.3 0.72 0.87 0.5 0.86 0.94 0.3 0.91 0.96 0.2 0.95 0.98 Fexviii 16.08 0.005 0.01 0.99 1.00 0.005 0.99 1.00 0.003 1.00 1.00 0.002 1.00 1.00
Figure 2.6: Observed radial profiles of the electron density and temperature used to model resonant scattering in NGC 4636. The blue circles and red squares indicate the deprojected profiles determined from Chandra data with metallicity as a free parameter in the fit and with metallicity fixed to 0.68 Solar, respectively. The magenta triangles show the projected radial temperature profile with the metallicity as a free parameter in the fit. The solid lines show the parametrization of the gas density and gas temperature profiles given in eqs. 2.3 and 2.4.
2.5 Modelling of resonant scattering in NGC 4636 35
the galaxy. The circles and squares show the deprojected profiles for metallicity as a free parameter and for metallicity fixed to 0.68 Solar, respectively. The triangles show the projected temperature profile with the metallicity as a free parameter.
Based on the observed radial distributions of electron densityne and temperature kTe
in NGC 4636, we adopt the following approximate forms for the deprojected density and temperature profiles: ne= 1.2×10−1 1 + r 0.25′ 2−0.75 cm−3, (2.3) and kT = 0.561 + 1.4(r/1.2 ′)4 1 + (r/1.2′)4 keV. (2.4)
We consider two possible radial behaviors for metallicity: a simple constant abundance of 0.68 Solar; and a centrally peaked abundance distribution, which is more consistent with the best fit XMM-Newton RGS line profile (s = 0.40±0.04). In the latter case we approximate the metallicity profile with the following function:
Z = 0.952 + (r/0.8
′)3
1 + (r/0.8′)3 −0.4. (2.5)
We calculate the optical depth of the 15 ˚A line from the centre of the galaxy to infinity,
τ = R
niσ0dr, where ni is the ion concentration and σ0 is the cross section at the line
centre, which for a given ion is
σ0 =
√
πhrecf
∆ED
, (2.6)
where the Doppler width is given by ∆ED =E0 2kTe Ampc2 + V 2 turb c2 1/2 . (2.7)
In these equations re is the classical electron radius,f is the oscillator strength of a given
atomic transition, E0 is the rest energy of a given line, A is the atomic mass of the
corresponding element, mp is the proton mass, c is the speed of light, and Vturb is the
characteristic velocity of isotropic turbulence. Vturb ≡
√
2V1D,turb, where V1D,turb is the
velocity dispersion in the line of sight due to turbulence. Using the adiabatic sound speed,
cs =
p
γ k T /µ mp, the expression for the broadening can be rewritten as
∆ED =E0 2kTe Ampc2 (1 + γ 2µAM 2) 1/2 , (2.8)
Figure 2.7: Optical depth of the 15.01 ˚A line calculated from the given radius to infinity, for isotropic turbulent velocities corresponding to Mach numbers 0.0, 0.25, 0.5, and 0.75. A centrally peaked Fe abundance distribution is assumed.
whereµ= 0.6 is the mean particle mass,γis the adiabatic index, which for ideal monatomic gas is 5/3, and M = Vturb/cs is the corresponding Mach number. The line energies and
oscillator strengths were taken from ATOMDB2 and the NIST Atomic Spectra Database3.
The resonant scattering effect has been calculated using Monte-Carlo simulations, as described by Churazov et al. (2004). We model the hot halo as a set of spherical shells and obtain line emissivities for each shell using the APEC plasma model (Smith et al., 2001). We account for scattering by assuming a complete energy redistribution and dipole scattering phase matrix. The line of Fexvii has a pure dipole scattering phase matrix. In
Fig. 2.7 we show the optical depth of the 15.01 ˚A Fexvii line as a function of radius, for
various turbulent velocities, assuming the peaked Fe abundance distribution. For higher turbulent velocities, the optical depth becomes smaller. In Fig. 2.8 we show the ratio of the 15.01 ˚A line intensity calculated including the effect of resonant scattering to the line intensity without accounting for resonant scattering for various turbulent velocities, both for constant and peaked iron abundance distributions. The figure clearly shows the effect of the resonant scattering: in the core of the galaxy the intensity of the line is suppressed, whereas in the surrounding regions the intensity of the line rises. The effect is more prominent in the case of the centrally peaked Fe abundance distribution, where the central suppression is larger and the enhancement in the outer part is stronger.
2
http://cxc.harvard.edu/atomdb/WebGUIDE/index.html
3
2.5 Modelling of resonant scattering in NGC 4636 37
Figure 2.8: Radial profiles of the ratio of the 15.01 ˚A line intensity with and without the effect of resonant scattering, for isotropic turbulent velocities corresponding to Mach num- bers 0.0, 0.25, 0.5, and 0.75. The upper panel shows the profiles for a constant metallicity, and the lower panel for a centrally peaked Fe abundance distribution.
We compare the models of resonant scattering with the observations and determine the systematic uncertainties involved by simulating spectra corresponding to each of these models, which are then fit in the same way as the observed data. To this end, we multiply the surface brightness profile of NGC 4636 with the theoretical suppression profiles for the 15.01 ˚A line (Fig. 2.8) to determine the predicted RGS line profile for the four considered characteristic turbulent velocities, for both assumed abundance distributions. We simulate spectra with photon statistics comparable to that of the actual observation, using the best fit values for the core of NGC 4636 from Table 2.2 as input parameters, and convolving the 15.01 ˚A line through these predicted line profiles. By fitting the data simulated with no turbulence, we obtain (I/I0)15.01˚A= 0.69 and 0.71 for peaked and flat abundance profiles,
respectively. The ratios of (I/I0)15.01˚Afor isotropic turbulent velocity ofM = 0.25 are 0.76
and 0.78, and forM = 0.50 they are 0.80 and 0.82, for peaked and flat abundance profiles, respectively. The systematic uncertainty on these values due to our lack of knowledge about the actual Fe abundance distribution is ±0.02. We add this uncertainty in quadrature to the uncertainties associated with our modeling of the line broadening, which is at most 0.03, resulting in a total uncertainty of 0.04. Thus we conclude that the turbulent velocities in NGC 4636 are relatively small and that isotropic turbulence with a characteristic velocity of M > 0.25 can be ruled out at the 90% confidence level; turbulent gas motions at
M >0.5 can be ruled out at the 95% confidence level. (The sound speed in NGC 4636 is
cs ∼400 km s−1.)
oscillator strengths and optical depths for different Mach numbers. Clearly, Fexvii at
15.01 ˚A is the most optically thick line in the spectrum, but OviiiLαand Fexviiiat 14.2
˚
A also have relatively large optical depths. For M = 0, the depth of the Fexvii line at
17.05 ˚A line is τ = 0.5, meaning that our assumption that the 17.05 ˚A and 17.10 ˚A line blend is optically thin is not completely correct, and our observed values of (I/I0)15.01˚A
are biased slightly high, which further increases the significance of our upper limits on the turbulent velocities. In Table 2.3, we also list the predicted suppression for the strongest emission lines, (I/I0)circ =
R
IdA /R
I0dA, integrated within a circular region of radius
2 kpc, and within the RGS extraction region (I/I0)RGS. The suppression for the RGS
is determined in a simplified way, integrating within our 0.5′ wide, and for line emission
effectively about 3′ long, RGS extraction region. These predicted ratios in Table 2.3 were
determined assuming a flat abundance distribution. By adopting an optically thin plasma in the analysis, the best fitting O abundance within the 2 kpc radius can be underestimated by ∼10–25%, and the Fe abundance by∼10–20%.